Although the choice \(\psi =0\)  has the advantage (exploited in [1]})
that the geometric rays \(\mathcal {R}^{\rm geo}_\psi (\gamma )\)  become
straight lines in \((x,y)\)  coordinates, it does not give access
to the index \(\Omega _{(s,t)}(\gamma )\)  away from loci
where \(Z^{\rm LV}_{(s,t)}(\gamma )\)  is purely imaginary. In §REF , we generalize the
scattering diagram \(\mathcal {D}^{\rm LV}_0\) 
to a diagram \(\mathcal {D}^{\rm LV}_\psi \)  valid for any
\(\psi \in (-\frac{\pi }{2},\frac{\pi }{2})\) .
While the walls of marginal stability are by construction independent of \(\psi \) , it turns out that the \(\psi \) -dependence of the rays \(\mathcal {R}_\psi (\gamma )\)  can be absorbed by a linear coordinate transformation \((s,t)\mapsto (s+t\tan \psi ,t/\cos \psi )\)  which preserves the walls and the boundary
at \(t=0\) . Thus, the topology of the trees contributing to the index at large volume for any charge \(\gamma \)  is independent of \(\psi \) , the only change being in the location of the
vertices along the walls of marginal stability. As we shall see momentarily, this is no longer true
for the exact scattering diagram involving the exact central charge function (REF ).
